双曲線関数入りの級数を計算する

こんな級数が知られているんですが、より一般的に
$$\sum _{n=1}^{\mathrm{\infty }}\frac{n^{2s}}{{\sinh }^{2r}\pi n}$$
の形の級数をガンマ関数で書く方法を見つけた(と言っても既知の結果ですが)ので、書きます。特殊値も幾つか計算したので最後に掲載しておきます。
最終的に重要となるのは正規化されたアイゼンシュタイン級数

とその微分です。では計算方法を説明します。

解説

$k$を正整数として、以下の変形を考えます。
$$\begin{array}{rl}\frac{d^2}{d{x}^2}\frac{1}{{\sinh }^{2k}\frac{x}{2}}& =-k\frac{d}{dx}\frac{\cosh \frac{x}{2}}{{\sinh }^{2k+1}\frac{x}{2}}\\ & =-\frac{k}{2}\frac{1}{{\sinh }^2\frac{x}{2}}+\frac{k(2k+1)}{2}\frac{{\cosh }^2\frac{x}{2}}{{\sinh }^{2k+2}\frac{x}{2}}\\ & =k^2\frac{1}{{\sinh }^{2k}\frac{x}{2}}+\frac{k(2k+1)}{2}\frac{1}{{\sinh }^{2(k+1)}\frac{x}{2}}\end{array}$$
これを繰り返し適用することで、$r$を正整数とするとき、定数$a_{r,1},...,a_{r,r}$を用いて
$$\begin{array}{rl}\frac{d^{2(r-1)}}{d{x}^{2(r-1)}}\frac{1}{4{\sinh }^2\frac{x}{2}}& =\sum _{k=1}^r{a}_{r,k}\frac{1}{{\sinh }^{2k}\frac{x}{2}}\end{array}$$
と表せることが分かります。例えば
$$\begin{array}{rl}\frac{d^2}{d{x}^2}\frac{1}{4{\sinh }^2\frac{x}{2}}& =\frac{1}{4}\frac{1}{{\sinh }^2\frac{x}{2}}+\frac{3}{8}\frac{1}{{\sinh }^4\frac{x}{2}}\end{array}$$
となります。
以下$x>0$とします。
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}{e}^{-nx}& =\frac{1}{e^x-1}\end{array}$$
の両辺を$x$で微分して
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}n{e}^{-2nx}& =\frac{1}{4{\sinh }^2\frac{x}{2}}\end{array}$$
を得ます。さらに両辺を$2(r-1)$回微分して
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}{n}^{2r-1}{e}^{-nx}& =\sum _{k=1}^r{a}_{r,k}\frac{1}{{\sinh }^{2k}\frac{x}{2}}\end{array}$$
となります。ここで、
$$\begin{array}{rl}{T}_r(x)& :=\sum _{n=1}^{\mathrm{\infty }}{n}^{2r-1}{e}^{-nx}\end{array}$$
とおきます。$T_r$を${\mathrm{csch}}^{2k}\frac{x}{2}$の線形結合で書くことができたので、今度は${\mathrm{csch}}^{2r}\frac{x}{2}$を$T_k$の線形結合で書くことを考えます。そのために以下の行列を定義します。
$$\begin{array}{rl}A& =\left[\begin{array}{ccc}{a}_{1,1}& & O\\ ⋮& \ddots & \\ a_{r,1}& \cdots & a_{r,r}\end{array}\right]\\ S& =\left[\begin{array}{c}\frac{1}{{\sinh }^2\frac{x}{2}}\\ ⋮\\ \frac{1}{{\sinh }^{2r}\frac{x}{2}}\end{array}\right],T=\left[\begin{array}{c}{T}_1\\ ⋮\\ T_r\end{array}\right]\end{array}$$
上記より$T=AS$です。また$a_{k,k}\ne 0$ですから$|A|\ne 0$となり$A$は正則です。従って
$$\begin{array}{rl}C& =\left[\begin{array}{}0& \cdots & 0& 1\end{array}\right]A^{-1}\\ & =\left[\begin{array}{}{c}_1& \cdots & c_r\end{array}\right]\end{array}$$
とすれば
$$\begin{array}{rl}CT& =\left[\begin{array}{}0& \cdots & 0& 1\end{array}\right]A^{-1}AS\\ & =\left[\begin{array}{}0& \cdots & 0& 1\end{array}\right]S\\ & =\frac{1}{{\sinh }^{2r}\frac{x}{2}}\end{array}$$
となり、
$$\begin{array}{rl}\frac{1}{{\sinh }^{2r}\frac{x}{2}}& =\sum _{k=1}^r{c}_k\sum _{n=1}^{\mathrm{\infty }}{n}^{2k-1}{e}^{-nx}\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{2s}}{{\sinh }^{2r}\frac{nx}{2}}& =\sum _{k=1}^r{c}_k\sum _{n,m=1}^{\mathrm{\infty }}{n}^{2k-1}{m}^{2s}{e}^{-nmx}\\ & =\sum _{k=1}^r{c}_k{\left(\frac{d}{dx}\right)}^{min(2k-1,2s)}\sum _{n,m=1}^{\mathrm{\infty }}{n}^{|2k-2s-1|}{e}^{-nmx}\\ & =\sum _{k=1}^r{c}_k{\left(\frac{d}{dx}\right)}^{min(2k-1,2s)}\sum _{n=1}^{\mathrm{\infty }}\frac{n^{|2k-2s-1|}}{e^{nx}-1}\\ & =\sum _{k=1}^r{c}_k\frac{\zeta (-1-2|2k-2s-1|)}{2}{\left(\frac{d}{dx}\right)}^{min(2k-1,2s)}{E}_{|2k-2s-1|+1}\left(\frac{ix}{2\pi }\right)\end{array}$$
最後の微分はこちらの記事の内容により
$E_2,E_4,E_6$で書けます。ここで$x=2\pi$とすれば
$$\sum _{n=1}^{\mathrm{\infty }}\frac{n^{2s}}{{\sinh }^{2r}\pi n}$$
を
$$\begin{array}{rl}{E}_2(i)=\frac{3}{\pi },E_4(i)=\frac{3{\pi }^2}{4\mathrm{\Gamma }{\left(\frac{3}{4}\right)}^8},E_6(i)& =0\end{array}$$
で書けることになります。アイゼンシュタイン級数の特殊値の求め方について補足すると、$E_2とE_6$の特殊値についてはモジュラー関係式
$$\begin{array}{rl}{E}_2(-\frac{1}{\tau })& ={\tau }^2{E}_2(\tau )-\frac{6\tau i}{\pi }\\ E_6(-\frac{1}{\tau })& ={\tau }^6{E}_6(\tau )\end{array}$$
において$\tau =i$とすれば求まり、$E_4$については
この記事で証明されている
$$\begin{array}{rl}\eta (\tau {)}^{24}& =\frac{E_4(\tau {)}^3-E_6(\tau {)}^2}{1728}\end{array}$$
において$\tau =i$とし、この記事で解説したイータ関数の特殊値
$$\begin{array}{rl}\eta (i)& =\frac{\sqrt[4]{\pi }}{\sqrt{2}\mathrm{\Gamma }\left(\frac{3}{4}\right)}\end{array}$$
を用いて求まります。

発展(Help!)

実は、途中出てきた$a_{r,k}$の明示式
$$\begin{array}{rl}{a}_{r,k}& =\frac{(2k-1)!}{2^{2k}}\sum _{1=n_0\le n_1\le \cdots \le n_{r-k}\le k}{n}_0^2\cdots n_{r-k}^2\end{array}$$
を得たんですが、使い道が見つからず困っています。うまく多重ゼータ値の話を持ち込めたりしないかな、という気持ちもあるので、何か知っている方は是非教えてください～

特殊値鑑賞会

プログラムを使って特殊値を大量に計算してみました。以下では定数$r$を
$$\begin{array}{rl}r& =\frac{{\pi }^4}{4\mathrm{\Gamma }{\left(\frac{3}{4}\right)}^8}\end{array}$$
として使います。
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^2\pi n}& =\frac{1}{6}-\frac{1}{2}{\pi }^{-1}\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^2\pi n}& =-\frac{1}{8}{\pi }^{-2}+\frac{1}{24}{\pi }^{-2}r\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^2\pi n}& =\frac{1}{20}{\pi }^{-3}r\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^2\pi n}& =\frac{1}{28}{\pi }^{-4}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^2\pi n}& =\frac{3}{20}{\pi }^{-5}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^2\pi n}& =\frac{9}{44}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^2\pi n}& =\frac{567}{260}{\pi }^{-7}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^2\pi n}& =-\frac{81}{8}{\pi }^{-6}{r}^2+\frac{3}{8}{\pi }^{-6}{r}^3\end{array}$$
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^4\pi n}& =-\frac{11}{90}+\frac{1}{3}{\pi }^{-1}+\frac{1}{30}{\pi }^{-2}r\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^4\pi n}& =\frac{1}{12}{\pi }^{-2}-\frac{1}{36}{\pi }^{-2}r-\frac{1}{24}{\pi }^{-3}+\frac{1}{24}{\pi }^{-3}r\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^4\pi n}& =-\frac{1}{30}{\pi }^{-3}r-\frac{1}{32}{\pi }^{-4}+\frac{1}{16}{\pi }^{-4}r+\frac{1}{96}{\pi }^{-4}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^4\pi n}& =-\frac{1}{42}{\pi }^{-4}{r}^2+\frac{1}{16}{\pi }^{-5}r+\frac{1}{16}{\pi }^{-5}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^4\pi n}& =-\frac{1}{10}{\pi }^{-5}{r}^2+\frac{1}{4}{\pi }^{-6}{r}^2+\frac{1}{72}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^4\pi n}& =-\frac{3}{22}{\pi }^{-6}{r}^3+\frac{9}{16}{\pi }^{-7}{r}^2+\frac{5}{16}{\pi }^{-7}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^4\pi n}& =-\frac{189}{130}{\pi }^{-7}{r}^3+\frac{27}{8}{\pi }^{-8}{r}^3+\frac{1}{4}{\pi }^{-8}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^4\pi n}& =\frac{27}{4}{\pi }^{-6}{r}^2-\frac{1}{4}{\pi }^{-6}{r}^3+\frac{1323}{80}{\pi }^{-9}{r}^3+\frac{171}{16}{\pi }^{-9}{r}^4\end{array}$$
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^6\pi n}& =\frac{191}{1890}-\frac{4}{15}{\pi }^{-1}-\frac{1}{30}{\pi }^{-2}r\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^6\pi n}& =-\frac{1}{15}{\pi }^{-2}+\frac{1}{45}{\pi }^{-2}r+\frac{1}{24}{\pi }^{-3}-\frac{1}{24}{\pi }^{-3}r+\frac{1}{120}{\pi }^{-4}r+\frac{1}{360}{\pi }^{-4}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^6\pi n}& =\frac{2}{75}{\pi }^{-3}r+\frac{1}{32}{\pi }^{-4}-\frac{1}{16}{\pi }^{-4}r-\frac{1}{96}{\pi }^{-4}{r}^2-\frac{1}{160}{\pi }^{-5}+\frac{1}{48}{\pi }^{-5}r+\frac{1}{96}{\pi }^{-5}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^6\pi n}& =\frac{2}{105}{\pi }^{-4}{r}^2-\frac{1}{16}{\pi }^{-5}r-\frac{1}{16}{\pi }^{-5}{r}^2-\frac{1}{128}{\pi }^{-6}+\frac{5}{128}{\pi }^{-6}r+\frac{5}{128}{\pi }^{-6}{r}^2+\frac{1}{1920}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^6\pi n}& =\frac{2}{25}{\pi }^{-5}{r}^2-\frac{1}{4}{\pi }^{-6}{r}^2-\frac{1}{72}{\pi }^{-6}{r}^3+\frac{7}{160}{\pi }^{-7}r+\frac{7}{48}{\pi }^{-7}{r}^2+\frac{1}{96}{\pi }^{-7}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^6\pi n}& =\frac{6}{55}{\pi }^{-6}{r}^3-\frac{9}{16}{\pi }^{-7}{r}^2-\frac{5}{16}{\pi }^{-7}{r}^3+\frac{15}{32}{\pi }^{-8}{r}^2+\frac{5}{32}{\pi }^{-8}{r}^3+\frac{1}{240}{\pi }^{-8}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^6\pi n}& =\frac{378}{325}{\pi }^{-7}{r}^3-\frac{27}{8}{\pi }^{-8}{r}^3-\frac{1}{4}{\pi }^{-8}{r}^4+\frac{297}{320}{\pi }^{-9}{r}^2+\frac{55}{32}{\pi }^{-9}{r}^3+\frac{13}{64}{\pi }^{-9}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^6\pi n}& =-\frac{27}{5}{\pi }^{-6}{r}^2+\frac{1}{5}{\pi }^{-6}{r}^3-\frac{1323}{80}{\pi }^{-9}{r}^3-\frac{171}{16}{\pi }^{-9}{r}^4+\frac{819}{64}{\pi }^{-10}{r}^3+\frac{91}{16}{\pi }^{-10}{r}^4+\frac{107}{960}{\pi }^{-10}{r}^5\end{array}$$
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^8\pi n}& =-\frac{2497}{28350}+\frac{8}{35}{\pi }^{-1}+\frac{7}{225}{\pi }^{-2}r+\frac{1}{1050}{\pi }^{-4}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^8\pi n}& =\frac{2}{35}{\pi }^{-2}-\frac{2}{105}{\pi }^{-2}r-\frac{7}{180}{\pi }^{-3}+\frac{7}{180}{\pi }^{-3}r-\frac{1}{90}{\pi }^{-4}r-\frac{1}{270}{\pi }^{-4}{r}^2+\frac{1}{630}{\pi }^{-5}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^8\pi n}& =-\frac{4}{175}{\pi }^{-3}r-\frac{7}{240}{\pi }^{-4}+\frac{7}{120}{\pi }^{-4}r+\frac{7}{720}{\pi }^{-4}{r}^2+\frac{1}{120}{\pi }^{-5}-\frac{1}{36}{\pi }^{-5}r-\frac{1}{72}{\pi }^{-5}{r}^2+\frac{1}{480}{\pi }^{-6}r+\frac{1}{240}{\pi }^{-6}{r}^2+\frac{1}{10080}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^8\pi n}& =-\frac{4}{245}{\pi }^{-4}{r}^2+\frac{7}{120}{\pi }^{-5}r+\frac{7}{120}{\pi }^{-5}{r}^2+\frac{1}{96}{\pi }^{-6}-\frac{5}{96}{\pi }^{-6}r-\frac{5}{96}{\pi }^{-6}{r}^2-\frac{1}{1440}{\pi }^{-6}{r}^3-\frac{1}{896}{\pi }^{-7}+\frac{1}{128}{\pi }^{-7}r+\frac{5}{384}{\pi }^{-7}{r}^2+\frac{1}{1920}{\pi }^{-7}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^8\pi n}& =-\frac{12}{175}{\pi }^{-5}{r}^2+\frac{7}{30}{\pi }^{-6}{r}^2+\frac{7}{540}{\pi }^{-6}{r}^3-\frac{7}{120}{\pi }^{-7}r-\frac{7}{36}{\pi }^{-7}{r}^2-\frac{1}{72}{\pi }^{-7}{r}^3-\frac{1}{512}{\pi }^{-8}+\frac{7}{384}{\pi }^{-8}r+\frac{35}{768}{\pi }^{-8}{r}^2+\frac{7}{1920}{\pi }^{-8}{r}^3+\frac{13}{161280}{\pi }^{-8}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^8\pi n}& =-\frac{36}{385}{\pi }^{-6}{r}^3+\frac{21}{40}{\pi }^{-7}{r}^2+\frac{7}{24}{\pi }^{-7}{r}^3-\frac{5}{8}{\pi }^{-8}{r}^2-\frac{5}{24}{\pi }^{-8}{r}^3-\frac{1}{180}{\pi }^{-8}{r}^4+\frac{3}{128}{\pi }^{-9}r+\frac{21}{128}{\pi }^{-9}{r}^2+\frac{5}{128}{\pi }^{-9}{r}^3+\frac{1}{384}{\pi }^{-9}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^8\pi n}& =-\frac{324}{325}{\pi }^{-7}{r}^3+\frac{63}{20}{\pi }^{-8}{r}^3+\frac{7}{30}{\pi }^{-8}{r}^4-\frac{99}{80}{\pi }^{-9}{r}^2-\frac{55}{24}{\pi }^{-9}{r}^3-\frac{13}{48}{\pi }^{-9}{r}^4+\frac{33}{64}{\pi }^{-10}{r}^2+\frac{55}{128}{\pi }^{-10}{r}^3+\frac{11}{160}{\pi }^{-10}{r}^4+\frac{1}{1920}{\pi }^{-10}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^8\pi n}& =\frac{162}{35}{\pi }^{-6}{r}^2-\frac{6}{35}{\pi }^{-6}{r}^3+\frac{3087}{200}{\pi }^{-9}{r}^3+\frac{399}{40}{\pi }^{-9}{r}^4-\frac{273}{16}{\pi }^{-10}{r}^3-\frac{91}{12}{\pi }^{-10}{r}^4-\frac{107}{720}{\pi }^{-10}{r}^5+\frac{1287}{1280}{\pi }^{-11}{r}^2+\frac{1001}{256}{\pi }^{-11}{r}^3+\frac{1183}{768}{\pi }^{-11}{r}^4+\frac{217}{3840}{\pi }^{-11}{r}^5\end{array}$$
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^{10}\pi n}& =\frac{14797}{187110}-\frac{64}{315}{\pi }^{-1}-\frac{82}{2835}{\pi }^{-2}r-\frac{1}{630}{\pi }^{-4}{r}^2\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^{10}\pi n}& =-\frac{16}{315}{\pi }^{-2}+\frac{16}{945}{\pi }^{-2}r+\frac{41}{1134}{\pi }^{-3}-\frac{41}{1134}{\pi }^{-3}r+\frac{13}{1080}{\pi }^{-4}r+\frac{13}{3240}{\pi }^{-4}{r}^2-\frac{1}{378}{\pi }^{-5}{r}^2+\frac{1}{4200}{\pi }^{-6}{r}^2+\frac{1}{22680}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^{10}\pi n}& =\frac{32}{1575}{\pi }^{-3}r+\frac{41}{1512}{\pi }^{-4}-\frac{41}{756}{\pi }^{-4}r-\frac{41}{4536}{\pi }^{-4}{r}^2-\frac{13}{1440}{\pi }^{-5}+\frac{13}{432}{\pi }^{-5}r+\frac{13}{864}{\pi }^{-5}{r}^2-\frac{1}{288}{\pi }^{-6}r-\frac{1}{144}{\pi }^{-6}{r}^2-\frac{1}{6048}{\pi }^{-6}{r}^3+\frac{1}{1260}{\pi }^{-7}{r}^2+\frac{1}{7560}{\pi }^{-7}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^{10}\pi n}& =\frac{32}{2205}{\pi }^{-4}{r}^2-\frac{41}{756}{\pi }^{-5}r-\frac{41}{756}{\pi }^{-5}{r}^2-\frac{13}{1152}{\pi }^{-6}+\frac{65}{1152}{\pi }^{-6}r+\frac{65}{1152}{\pi }^{-6}{r}^2+\frac{13}{17280}{\pi }^{-6}{r}^3+\frac{5}{2688}{\pi }^{-7}-\frac{5}{384}{\pi }^{-7}r-\frac{25}{1152}{\pi }^{-7}{r}^2-\frac{1}{1152}{\pi }^{-7}{r}^3+\frac{1}{1920}{\pi }^{-8}r+\frac{1}{384}{\pi }^{-8}{r}^2+\frac{1}{2688}{\pi }^{-8}{r}^3+\frac{1}{120960}{\pi }^{-8}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^{10}\pi n}& =\frac{32}{525}{\pi }^{-5}{r}^2-\frac{41}{189}{\pi }^{-6}{r}^2-\frac{41}{3402}{\pi }^{-6}{r}^3+\frac{91}{1440}{\pi }^{-7}r+\frac{91}{432}{\pi }^{-7}{r}^2+\frac{13}{864}{\pi }^{-7}{r}^3+\frac{5}{1536}{\pi }^{-8}-\frac{35}{1152}{\pi }^{-8}r-\frac{175}{2304}{\pi }^{-8}{r}^2-\frac{7}{1152}{\pi }^{-8}{r}^3-\frac{13}{96768}{\pi }^{-8}{r}^4-\frac{1}{4608}{\pi }^{-9}+\frac{1}{384}{\pi }^{-9}r+\frac{7}{768}{\pi }^{-9}{r}^2+\frac{7}{5760}{\pi }^{-9}{r}^3+\frac{13}{161280}{\pi }^{-9}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^{10}\pi n}& =\frac{32}{385}{\pi }^{-6}{r}^3-\frac{41}{84}{\pi }^{-7}{r}^2-\frac{205}{756}{\pi }^{-7}{r}^3+\frac{65}{96}{\pi }^{-8}{r}^2+\frac{65}{288}{\pi }^{-8}{r}^3+\frac{13}{2160}{\pi }^{-8}{r}^4-\frac{5}{128}{\pi }^{-9}r-\frac{35}{128}{\pi }^{-9}{r}^2-\frac{25}{384}{\pi }^{-9}{r}^3-\frac{5}{1152}{\pi }^{-9}{r}^4-\frac{1}{2048}{\pi }^{-10}+\frac{15}{2048}{\pi }^{-10}r+\frac{35}{1024}{\pi }^{-10}{r}^2+\frac{7}{1024}{\pi }^{-10}{r}^3+\frac{13}{14336}{\pi }^{-10}{r}^4+\frac{1}{645120}{\pi }^{-10}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^{10}\pi n}& =\frac{288}{325}{\pi }^{-7}{r}^3-\frac{41}{14}{\pi }^{-8}{r}^3-\frac{41}{189}{\pi }^{-8}{r}^4+\frac{429}{320}{\pi }^{-9}{r}^2+\frac{715}{288}{\pi }^{-9}{r}^3+\frac{169}{576}{\pi }^{-9}{r}^4-\frac{55}{64}{\pi }^{-10}{r}^2-\frac{275}{384}{\pi }^{-10}{r}^3-\frac{11}{96}{\pi }^{-10}{r}^4-\frac{1}{1152}{\pi }^{-10}{r}^5+\frac{11}{1024}{\pi }^{-11}r+\frac{33}{256}{\pi }^{-11}{r}^2+\frac{33}{512}{\pi }^{-11}{r}^3+\frac{11}{768}{\pi }^{-11}{r}^4+\frac{19}{107520}{\pi }^{-11}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^{10}\pi n}& =-\frac{144}{35}{\pi }^{-6}{r}^2+\frac{16}{105}{\pi }^{-6}{r}^3-\frac{287}{20}{\pi }^{-9}{r}^3-\frac{779}{84}{\pi }^{-9}{r}^4+\frac{1183}{64}{\pi }^{-10}{r}^3+\frac{1183}{144}{\pi }^{-10}{r}^4+\frac{1391}{8640}{\pi }^{-10}{r}^5-\frac{429}{256}{\pi }^{-11}{r}^2-\frac{5005}{768}{\pi }^{-11}{r}^3-\frac{5915}{2304}{\pi }^{-11}{r}^4-\frac{217}{2304}{\pi }^{-11}{r}^5+\frac{429}{1024}{\pi }^{-12}{r}^2+\frac{1001}{1536}{\pi }^{-12}{r}^3+\frac{1001}{3840}{\pi }^{-12}{r}^4+\frac{91}{7680}{\pi }^{-12}{r}^5+\frac{29}{322560}{\pi }^{-12}{r}^6\end{array}$$
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^{12}\pi n}& =-\frac{92427157}{1277025750}+\frac{128}{693}{\pi }^{-1}+\frac{1916}{70875}{\pi }^{-2}r+\frac{31}{15750}{\pi }^{-4}{r}^2+\frac{1}{53625}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^{12}\pi n}& =\frac{32}{693}{\pi }^{-2}-\frac{32}{2079}{\pi }^{-2}r-\frac{479}{14175}{\pi }^{-3}+\frac{479}{14175}{\pi }^{-3}r-\frac{139}{11340}{\pi }^{-4}r-\frac{139}{34020}{\pi }^{-4}{r}^2+\frac{31}{9450}{\pi }^{-5}{r}^2-\frac{1}{2100}{\pi }^{-6}{r}^2-\frac{1}{11340}{\pi }^{-6}{r}^3+\frac{1}{34650}{\pi }^{-7}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^{12}\pi n}& =-\frac{64}{3465}{\pi }^{-3}r-\frac{479}{18900}{\pi }^{-4}+\frac{479}{9450}{\pi }^{-4}r+\frac{479}{56700}{\pi }^{-4}{r}^2+\frac{139}{15120}{\pi }^{-5}-\frac{139}{4536}{\pi }^{-5}r-\frac{139}{9072}{\pi }^{-5}{r}^2+\frac{31}{7200}{\pi }^{-6}r+\frac{31}{3600}{\pi }^{-6}{r}^2+\frac{31}{151200}{\pi }^{-6}{r}^3-\frac{1}{630}{\pi }^{-7}{r}^2-\frac{1}{3780}{\pi }^{-7}{r}^3+\frac{1}{16800}{\pi }^{-8}{r}^2+\frac{1}{15120}{\pi }^{-8}{r}^3+\frac{13}{4989600}{\pi }^{-8}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^{12}\pi n}& =-\frac{64}{4851}{\pi }^{-4}{r}^2+\frac{479}{9450}{\pi }^{-5}r+\frac{479}{9450}{\pi }^{-5}{r}^2+\frac{139}{12096}{\pi }^{-6}-\frac{695}{12096}{\pi }^{-6}r-\frac{695}{12096}{\pi }^{-6}{r}^2-\frac{139}{181440}{\pi }^{-6}{r}^3-\frac{31}{13440}{\pi }^{-7}+\frac{31}{1920}{\pi }^{-7}r+\frac{31}{1152}{\pi }^{-7}{r}^2+\frac{31}{28800}{\pi }^{-7}{r}^3-\frac{1}{960}{\pi }^{-8}r-\frac{1}{192}{\pi }^{-8}{r}^2-\frac{1}{1344}{\pi }^{-8}{r}^3-\frac{1}{60480}{\pi }^{-8}{r}^4+\frac{1}{3360}{\pi }^{-9}{r}^2+\frac{1}{6048}{\pi }^{-9}{r}^3+\frac{1}{75600}{\pi }^{-9}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^{12}\pi n}& =-\frac{64}{1155}{\pi }^{-5}{r}^2+\frac{958}{4725}{\pi }^{-6}{r}^2+\frac{479}{42525}{\pi }^{-6}{r}^3-\frac{139}{2160}{\pi }^{-7}r-\frac{139}{648}{\pi }^{-7}{r}^2-\frac{139}{9072}{\pi }^{-7}{r}^3-\frac{31}{7680}{\pi }^{-8}+\frac{217}{5760}{\pi }^{-8}r+\frac{217}{2304}{\pi }^{-8}{r}^2+\frac{217}{28800}{\pi }^{-8}{r}^3+\frac{403}{2419200}{\pi }^{-8}{r}^4+\frac{1}{2304}{\pi }^{-9}-\frac{1}{192}{\pi }^{-9}r-\frac{7}{384}{\pi }^{-9}{r}^2-\frac{7}{2880}{\pi }^{-9}{r}^3-\frac{13}{80640}{\pi }^{-9}{r}^4+\frac{1}{7680}{\pi }^{-10}r+\frac{7}{5760}{\pi }^{-10}{r}^2+\frac{1}{2304}{\pi }^{-10}{r}^3+\frac{1}{17280}{\pi }^{-10}{r}^4+\frac{19}{79833600}{\pi }^{-10}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^{12}\pi n}& =-\frac{64}{847}{\pi }^{-6}{r}^3+\frac{479}{1050}{\pi }^{-7}{r}^2+\frac{479}{1890}{\pi }^{-7}{r}^3-\frac{695}{1008}{\pi }^{-8}{r}^2-\frac{695}{3024}{\pi }^{-8}{r}^3-\frac{139}{22680}{\pi }^{-8}{r}^4+\frac{31}{640}{\pi }^{-9}r+\frac{217}{640}{\pi }^{-9}{r}^2+\frac{31}{384}{\pi }^{-9}{r}^3+\frac{31}{5760}{\pi }^{-9}{r}^4+\frac{1}{1024}{\pi }^{-10}-\frac{15}{1024}{\pi }^{-10}r-\frac{35}{512}{\pi }^{-10}{r}^2-\frac{7}{512}{\pi }^{-10}{r}^3-\frac{13}{7168}{\pi }^{-10}{r}^4-\frac{1}{322560}{\pi }^{-10}{r}^5-\frac{1}{22528}{\pi }^{-11}+\frac{5}{6144}{\pi }^{-11}r+\frac{5}{1024}{\pi }^{-11}{r}^2+\frac{7}{5120}{\pi }^{-11}{r}^3+\frac{13}{43008}{\pi }^{-11}{r}^4+\frac{1}{645120}{\pi }^{-11}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^{12}\pi n}& =-\frac{576}{715}{\pi }^{-7}{r}^3+\frac{479}{175}{\pi }^{-8}{r}^3+\frac{958}{4725}{\pi }^{-8}{r}^4-\frac{1529}{1120}{\pi }^{-9}{r}^2-\frac{7645}{3024}{\pi }^{-9}{r}^3-\frac{1807}{6048}{\pi }^{-9}{r}^4+\frac{341}{320}{\pi }^{-10}{r}^2+\frac{341}{384}{\pi }^{-10}{r}^3+\frac{341}{2400}{\pi }^{-10}{r}^4+\frac{31}{28800}{\pi }^{-10}{r}^5-\frac{11}{512}{\pi }^{-11}r-\frac{33}{128}{\pi }^{-11}{r}^2-\frac{33}{256}{\pi }^{-11}{r}^3-\frac{11}{384}{\pi }^{-11}{r}^4-\frac{19}{53760}{\pi }^{-11}{r}^5-\frac{1}{8192}{\pi }^{-12}+\frac{11}{4096}{\pi }^{-12}r+\frac{165}{8192}{\pi }^{-12}{r}^2+\frac{77}{10240}{\pi }^{-12}{r}^3+\frac{143}{57344}{\pi }^{-12}{r}^4+\frac{11}{430080}{\pi }^{-12}{r}^5+\frac{149}{425779200}{\pi }^{-12}{r}^6\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^{12}\pi n}& =\frac{288}{77}{\pi }^{-6}{r}^2-\frac{32}{231}{\pi }^{-6}{r}^3+\frac{3353}{250}{\pi }^{-9}{r}^3+\frac{9101}{1050}{\pi }^{-9}{r}^4-\frac{1807}{96}{\pi }^{-10}{r}^3-\frac{1807}{216}{\pi }^{-10}{r}^4-\frac{14873}{90720}{\pi }^{-10}{r}^5+\frac{13299}{6400}{\pi }^{-11}{r}^2+\frac{31031}{3840}{\pi }^{-11}{r}^3+\frac{36673}{11520}{\pi }^{-11}{r}^4+\frac{6727}{57600}{\pi }^{-11}{r}^5-\frac{429}{512}{\pi }^{-12}{r}^2-\frac{1001}{768}{\pi }^{-12}{r}^3-\frac{1001}{1920}{\pi }^{-12}{r}^4-\frac{91}{3840}{\pi }^{-12}{r}^5-\frac{29}{161280}{\pi }^{-12}{r}^6+\frac{91}{20480}{\pi }^{-13}r+\frac{1001}{12288}{\pi }^{-13}{r}^2+\frac{143}{2048}{\pi }^{-13}{r}^3+\frac{1001}{30720}{\pi }^{-13}{r}^4+\frac{247}{184320}{\pi }^{-13}{r}^5+\frac{31}{921600}{\pi }^{-13}{r}^6\end{array}$$
$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\sinh }^{14}\pi n}& =\frac{8113594822600175}{120861989803434770}-\frac{2379130053706055}{13954155373592348}{\pi }^{-1}-\frac{887052031829709}{34885388433980870}{\pi }^{-2}r-\frac{4898474577706915}{2232664859774775800}{\pi }^{-4}{r}^2-\frac{169182581596875}{57156220410234260000}{\pi }^{-5}{r}^2-\frac{3991455721378125}{91732205596672260000}{\pi }^{-6}{r}^3\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{{\sinh }^{14}\pi n}& =-\frac{7137390161118165}{167449864483108200}{\pi }^{-2}+\frac{2379130053706055}{167449864483108200}{\pi }^{-2}r+\frac{7983468286467381}{251174796724662270}{\pi }^{-3}-\frac{2661156095489127}{83724932241554100}{\pi }^{-3}r+\frac{6119155541604943}{502349593449324540}{\pi }^{-4}r+\frac{8158874055473257}{2009398373797298200}{\pi }^{-4}{r}^2-\frac{95424829435849}{26096082776588290}{\pi }^{-5}{r}^2+\frac{2976886479699701}{4465329719549551600}{\pi }^{-6}{r}^2+\frac{360834724812085}{2922761270977888000}{\pi }^{-6}{r}^3-\frac{5773355596993359}{85734330615351400000}{\pi }^{-7}{r}^3+\frac{2664625660150781}{571562204102342600000}{\pi }^{-8}{r}^3+\frac{1928681430204375}{1.9204490057838714e+21}{\pi }^{-8}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{\sinh }^{14}\pi n}& =\frac{7137390161118165}{418624661207770500}{\pi }^{-3}r+\frac{748450151856317}{31396849590582784}{\pi }^{-4}-\frac{7983468286467381}{167449864483108200}{\pi }^{-4}r-\frac{2661156095489127}{334899728966216400}{\pi }^{-4}{r}^2-\frac{78227840730745}{8562777161068032}{\pi }^{-5}+\frac{8741650773721347}{287056910542471170}{\pi }^{-5}r+\frac{8741650773721347}{574113821084942340}{\pi }^{-5}{r}^2-\frac{657536715331397}{137004434577088510}{\pi }^{-6}r-\frac{688847987490035}{71764227635617790}{\pi }^{-6}{r}^2-\frac{95424829435849}{417537324425412600}{\pi }^{-6}{r}^3+\frac{8930659439099103}{4018796747594596400}{\pi }^{-7}{r}^2+\frac{2976886479699701}{8037593495189193000}{\pi }^{-7}{r}^3-\frac{7422885767562891}{53444777526452810000}{\pi }^{-8}{r}^2-\frac{620184683270771}{4018796747594596400}{\pi }^{-8}{r}^3-\frac{2599259662715625}{427558220211622500000}{\pi }^{-8}{r}^4+\frac{771208910915625}{53444777526452810000}{\pi }^{-9}{r}^3+\frac{85689878990625}{26722388763226407000}{\pi }^{-9}{r}^4\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^6}{{\sinh }^{14}\pi n}& =\frac{1427478032223633}{117214905138175740}{\pi }^{-4}{r}^2-\frac{748450151856317}{15698424795291392}{\pi }^{-5}r-\frac{7983468286467381}{167449864483108200}{\pi }^{-5}{r}^2-\frac{2458589280109129}{215292682906853380}{\pi }^{-6}+\frac{1792721350079573}{31396849590582784}{\pi }^{-6}r+\frac{1792721350079573}{31396849590582784}{\pi }^{-6}{r}^2+\frac{6119155541604943}{8037593495189193000}{\pi }^{-6}{r}^3+\frac{2906077447223585}{1130286585260980200}{\pi }^{-7}-\frac{352251811784677}{19572062082441216}{\pi }^{-7}r-\frac{5650706147379193}{188381097543496700}{\pi }^{-7}{r}^2-\frac{688847987490035}{574113821084942340}{\pi }^{-7}{r}^3+\frac{2930372628454393}{2009398373797298200}{\pi }^{-8}r+\frac{457870723195999}{62793699181165570}{\pi }^{-8}{r}^2+\frac{1395415537359235}{1339598915864865500}{\pi }^{-8}{r}^3+\frac{2976886479699701}{128601495923027080000}{\pi }^{-8}{r}^4-\frac{8372493224155409}{12056390242783790000}{\pi }^{-9}{r}^2-\frac{4651385124530783}{12056390242783790000}{\pi }^{-9}{r}^3-\frac{6598120682278125}{213779110105811260000}{\pi }^{-9}{r}^4+\frac{3181236757526953}{213779110105811260000}{\pi }^{-10}{r}^2+\frac{5980352302968149}{144676682913405470000}{\pi }^{-10}{r}^3+\frac{4177381600792969}{427558220211622500000}{\pi }^{-10}{r}^4+\frac{204337403746875}{1.71023288084649e+21}{\pi }^{-10}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^8}{{\sinh }^{14}\pi n}& =\frac{5353042620838623}{104656165301942620}{\pi }^{-5}{r}^2-\frac{748450151856317}{3924606198822848}{\pi }^{-6}{r}^2-\frac{2661156095489127}{251174796724662270}{\pi }^{-6}{r}^3+\frac{8031391648356487}{125587398362331140}{\pi }^{-7}r+\frac{7529429670334207}{35321455789405630}{\pi }^{-7}{r}^2+\frac{8741650773721347}{574113821084942340}{\pi }^{-7}{r}^3+\frac{5547966035608663}{1233039911193796600}{\pi }^{-8}-\frac{8899862182122229}{211928734736433800}{\pi }^{-8}r-\frac{3708275909217595}{35321455789405630}{\pi }^{-8}{r}^2-\frac{263699620211029}{31396849590582784}{\pi }^{-8}{r}^3-\frac{5970015891580303}{32150373980756770000}{\pi }^{-8}{r}^4-\frac{3090627381572993}{5086289633674411000}{\pi }^{-9}+\frac{457870723195999}{62793699181165570}{\pi }^{-9}r+\frac{4807642593557989}{188381097543496700}{\pi }^{-9}{r}^2+\frac{1282038024948797}{376762195086993400}{\pi }^{-9}{r}^3+\frac{518297199590573}{2296455284339769300}{\pi }^{-9}{r}^4-\frac{2996972006373811}{9864319289550373000}{\pi }^{-10}r-\frac{6410190124743985}{2260573170521960400}{\pi }^{-10}{r}^2-\frac{3329968895970901}{3288106429850124300}{\pi }^{-10}{r}^3-\frac{3551966822368961}{26304851438800994000}{\pi }^{-10}{r}^4-\frac{2570947414286105}{4.629653853228975e+21}{\pi }^{-10}{r}^5+\frac{2651030631272461}{26722388763226407000}{\pi }^{-11}{r}^2+\frac{523280826509713}{4521146341043921000}{\pi }^{-11}{r}^3+\frac{6598120682278125}{213779110105811260000}{\pi }^{-11}{r}^4+\frac{599829152934375}{855116440423245000000}{\pi }^{-11}{r}^5\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{10}}{{\sinh }^{14}\pi n}& =\frac{5353042620838623}{76747854554757920}{\pi }^{-6}{r}^3-\frac{2245350455568951}{5232808265097131}{\pi }^{-7}{r}^2-\frac{3742250759281585}{15698424795291392}{\pi }^{-7}{r}^3+\frac{576231862525577}{840987042604896}{\pi }^{-8}{r}^2+\frac{1792721350079573}{7849212397645696}{\pi }^{-8}{r}^3+\frac{655623808029101}{107646341453426690}{\pi }^{-8}{r}^4-\frac{5547966035608663}{102753325932816380}{\pi }^{-9}r-\frac{8899862182122229}{23547637192937090}{\pi }^{-9}{r}^2-\frac{1210865603009827}{13455792681678336}{\pi }^{-9}{r}^3-\frac{3287683576656985}{548017738308354050}{\pi }^{-9}{r}^4-\frac{2317970536179745}{1695429877891470300}{\pi }^{-10}+\frac{21}{1024}{\pi }^{-10}r+\frac{4507164931460615}{47095274385874180}{\pi }^{-10}{r}^2+\frac{4807642593557989}{251174796724662270}{\pi }^{-10}{r}^3+\frac{6802650744626269}{2679197831729731000}{\pi }^{-10}{r}^4+\frac{8930659439099103}{2.0576239347684333e+21}{\pi }^{-10}{r}^5+\frac{8428983767926343}{81380634138790580000}{\pi }^{-11}-\frac{35}{18432}{\pi }^{-11}r-\frac{35}{3072}{\pi }^{-11}{r}^2-\frac{6410190124743985}{2009398373797298200}{\pi }^{-11}{r}^3-\frac{8245637266213661}{11691045083911553000}{\pi }^{-11}{r}^4-\frac{1860554049812313}{514405983692108330000}{\pi }^{-11}{r}^5+\frac{1324554592102711}{40690317069395290000}{\pi }^{-12}r+\frac{1}{2048}{\pi }^{-12}{r}^2+\frac{7849212397645697}{24112780485567580000}{\pi }^{-12}{r}^3+\frac{5232808265097131}{48225560971135160000}{\pi }^{-12}{r}^4+\frac{4579052908561523}{1.71023288084649e+21}{\pi }^{-12}{r}^5+\frac{613012211240625}{2.736372609354384e+22}{\pi }^{-12}{r}^6\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{12}}{{\sinh }^{14}\pi n}& =\frac{7025868439850693}{9448126034203152}{\pi }^{-7}{r}^3-\frac{8981401822275804}{3488538843398087}{\pi }^{-8}{r}^3-\frac{748450151856317}{3924606198822848}{\pi }^{-8}{r}^4+\frac{1331095602434083}{981151549705712}{\pi }^{-9}{r}^2+\frac{2464991856359413}{981151549705712}{\pi }^{-9}{r}^3+\frac{2330537755103445}{7849212397645696}{\pi }^{-9}{r}^4-\frac{5244561643036313}{4415181973675704}{\pi }^{-10}{r}^2-\frac{4994820612415537}{5045922255629376}{\pi }^{-10}{r}^3-\frac{3729466057270267}{23547637192937090}{\pi }^{-10}{r}^4-\frac{657536715331397}{548017738308354050}{\pi }^{-10}{r}^5+\frac{8499225299325731}{282571646315245060}{\pi }^{-11}r+\frac{5666150199550487}{15698424795291392}{\pi }^{-11}{r}^2+\frac{5666150199550487}{31396849590582784}{\pi }^{-11}{r}^3+\frac{457870723195999}{11417036214757376}{\pi }^{-11}{r}^4+\frac{7953868562947639}{16075186990378385000}{\pi }^{-11}{r}^5+\frac{4346194755337021}{15258868901023234000}{\pi }^{-12}-\frac{1770671937359527}{282571646315245060}{\pi }^{-12}r-\frac{3320009882549113}{70642911578811260}{\pi }^{-12}{r}^2-\frac{1239470356151669}{70642911578811260}{\pi }^{-12}{r}^3-\frac{182688374489475}{31396849590582784}{\pi }^{-12}{r}^4-\frac{479674090967237}{8037593495189193000}{\pi }^{-12}{r}^5-\frac{8640702963803677}{1.0582065950237657e+22}{\pi }^{-12}{r}^6-\frac{4584996664970923}{488283804832743500000}{\pi }^{-13}+\frac{1}{4096}{\pi }^{-13}r+\frac{55}{24576}{\pi }^{-13}{r}^2+\frac{2428350085521637}{2260573170521960400}{\pi }^{-13}{r}^3+\frac{429504096758929}{861170731627413500}{\pi }^{-13}{r}^4+\frac{1644596883316241}{192902243884540620000}{\pi }^{-13}{r}^5+\frac{720058580316973}{2.0576239347684333e+21}{\pi }^{-13}{r}^6\\ \sum _{n=1}^{\mathrm{\infty }}\frac{n^{14}}{{\sinh }^{14}\pi n}& =-\frac{16059127862515870}{4651385124530783}{\pi }^{-6}{r}^2+\frac{5353042620838623}{41862466120777050}{\pi }^{-6}{r}^3-\frac{41258314621079470}{3270505165685707}{\pi }^{-9}{r}^3-\frac{42661658655810060}{5232808265097131}{\pi }^{-9}{r}^4+\frac{5243709948982751}{280329014201632}{\pi }^{-10}{r}^3+\frac{3495806632655167}{420493521302448}{\pi }^{-10}{r}^4+\frac{1918211844585143}{11773818596468544}{\pi }^{-10}{r}^5-\frac{2272643378649069}{981151549705712}{\pi }^{-11}{r}^2-\frac{1657135796931613}{183965915569821}{\pi }^{-11}{r}^3-\frac{1627788645864347}{458720205057216}{\pi }^{-11}{r}^4-\frac{766377021238303}{5886909298234272}{\pi }^{-11}{r}^5+\frac{1726405138925539}{1471727324558568}{\pi }^{-12}{r}^2+\frac{651034934544311}{356782381711168}{\pi }^{-12}{r}^3+\frac{6445245851988679}{8830363947351408}{\pi }^{-12}{r}^4+\frac{8333247162167181}{251174796724662270}{\pi }^{-12}{r}^5+\frac{367882278031071}{1461380635488944000}{\pi }^{-12}{r}^6-\frac{2471898267097931}{238419826578488030}{\pi }^{-13}r-\frac{5664766862099425}{29802478322311004}{\pi }^{-13}{r}^2-\frac{8632025694627695}{52982183684108450}{\pi }^{-13}{r}^3-\frac{651034934544311}{8562777161068032}{\pi }^{-13}{r}^4-\frac{504884234952731}{161469512180140030}{\pi }^{-13}{r}^5-\frac{1081447041453407}{13778731706038616000}{\pi }^{-13}{r}^6-\frac{1}{32768}{\pi }^{-14}+\frac{2648462429033497}{2861037918941856300}{\pi }^{-14}r+\frac{2427757226614039}{238419826578488030}{\pi }^{-14}{r}^2+\frac{109613024693685}{17941056908904448}{\pi }^{-14}{r}^3+\frac{474989773672635}{125587398362331140}{\pi }^{-14}{r}^4+\frac{12179224965965}{125587398362331140}{\pi }^{-14}{r}^5+\frac{1727718683436331}{217015024370108200000}{\pi }^{-14}{r}^6+\frac{4475648294971875}{4.3781961749670145e+23}{\pi }^{-14}{r}^7\end{array}$$